Spring Solution Homework.pdf

# X t follows sarima 0 1 2 1 1 4 implies that y t 1 b x

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X t follows SARIMA (0 , 1 , 2) × (1 , 0 , 1) 4 implies that Y t = (1 - B ) X t follows the multiplicative model (1 - Φ 1 B 4 ) Y t = (1 + θ 1 B + θ 2 B 2 )(1 + Θ 1 B 4 ) Z t (1 - Φ 1 B 4 )(1 - B ) X t = (1 + θ 1 B + θ 2 B 2 + Θ 1 B 4 + θ 1 Θ 1 B 5 + θ 2 Θ 1 B 6 ) Z t X t - X t - 1 - Φ 1 X t - 4 + Φ 1 X t - 5 = Z t + θ 1 Z t - 1 + θ 2 Z t - 2 + Θ 1 Z t - 4 + θ 1 Θ 1 Z t - 5 + θ 2 Θ 1 Z t - 6 . It is nonstationary ARMA(5, 6) model with 7 nonzero coefficients. Only four coefficients Φ 1 , Θ 1 , θ 1 , θ 2 , must be estimated from data. (c) We are given that ρ X (12) 6 = 0 , ρ X (24) 6 = 0 , ρ X ( k ) = 0 , k 6 = 12 , 24 . This implies pure SMA with Q=2, s=12: X t = (1 + Θ 1 B 12 + Θ 2 B 24 ) Z t = Z t + Θ 1 Z t - 1 + Θ 2 Z t - 24 . X t follows SARIMA (0 , 0 , 0) × (0 , 0 , 2) 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Analyze in R the deaths.txt dataset posted on Gaucho Space. The dataset is also available at ; search “accidental deaths”. (a) Plot the time series and find its mean, variance, acf and pacf. Notice the presence of a strong seasonal component with period 12 in the graphs of the data and in he sample acf. Comment on all plots. (b) Use differencing to deseasonalize and detrend the data. On each step include graphs of the data, corresponding variance, acf and pacf, and comment on your decisions to difference (or not) and at what lag. Please include the code with clear comments explaining the meaning of the code. Make sure to label the graphs. Solution. After you’ve set your working directory using setwd() we read the data using read.table (here is recommendable to open the raw file to have a quick glance about the structure of the data, e.g. no column names are included) deaths < - read . table ( ” deaths . t x t ” , header = FALSE) we now see whether the data was properly input head ( deaths ) ## V1 ## 1 9007 ## 2 8106 ## 3 8928 ## 4 9137 ## 5 10017 ## 6 10826 To perform exploratory data analysis, we first convert the raw data into a ts object (this is not required for this question but it always a good practice for further analysis): deaths < - ts ( deaths ) This study resource was shared via CourseHero.com

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Summary statistics mean ( deaths ) ## [1] 8787.736 var ( deaths ) ## V1 ## V1 918411.7 Plot of time series: plot ( deaths , ylab = ” Deaths ” , xlab = ”Time” , col = ” blue ” ) time deaths 0 10 20 30 40 50 60 70 7000 8000 9000 10000 11000 In general, when we analyze the plot of the raw time series data there are three factors that we need to look for: constant variance, trend and seasonal components. Constant variance: since the range of values that the series can assume is constant across all time, we can conclude that the variance does not vary with time (therefore no Box-Cox transformation is needed).
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• Spring '17
• Sams
• ........., sh, ed d, ACF

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